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120 VI HUBERT SPACES

such that \\x — y\\ = d(x, F). The point y = P_F(x) is also the only point z e F
such that x - z is orthogonal to F. The mapping x -> P_F(x) 0/ E onto F w
/we<2r, continuous, and of norm 1 z/ F ^ {0}; its kernel F' = P~¹(Q) is the
subspace orthogonal to ¥,andE is the topological direct sum (see Section (5.4))
o/F and F'. Finally, F w rte subspace orthogonal to F'.

Let a = rf(je, F); by definition, there exists a sequence (y_n) of points
of F such that lim ||x — j>J = a; we prove (y_n) is a Cauchy sequence.

H-+OO

Indeed, for any two points u, v of E, it follows from (6.1 .1) that
(6.3.1 .1) \\u + v\\² + \\u - v\\² = 2(N|² + ||i?||²)

hence \\y_m - yj² = 2(||* - y_a\\² + ||x - jj²) - 4||* - Ky_M + 7«)ll². But
iC^m + 7J e F, hence \\x - %(y_m + y_n)\\² ^ a² ; therefore, if n_Q is such that for
n ^ «₀ , \\x - y_n\\² < a² + e, we have, for m ^ w₀ and n ^ «₀ > Ibm - ^nii^? < ^4fi₃which proves our contention. As F is complete, the sequence (y_n) tends
to a limit y e F, for which ||x - y\\ = d(x₉ F). Suppose y' e F is also such
that \\x—y'\\=d(x,]?); using again (6.3.11), we obtain ||j>-/||² =
4a²- 4 \\x - $(y + /)||², and as %(y + /) e F, this implies ||j; - / 1|² < 0, i.e.
y^r - y. Let now z ^ 0 be any point of F, and write that \\x - ( y + Az)||² > a²for any real scalar A ^ 0; this, by (6.1 .1 ), gives

and this would yield a contradiction if we had $(x — y \ z) ^ 0, by a suitable
choice of L Hence 3%(x — y \ z) = 0, and replacing z by /z (if E is a complex
prehilbert space) shows that J(x — y \ z) = 0, hence (x — y \ z) = 0 in
every case; in other words x — y is orthogonal to F. Let y' e F be such
that x — / is orthogonal to F; then, for any z^O in F, we have
\\x - (/ + z)||² = ||x - / 1|² + ||z||² by Pythagoras' theorem, and this proves
that / = y by the previous characterization of y.

This last characterization of y = P_F(x) proves that P_F is linear, for if
x — y and x' — y' are orthogonal to F, then Xx — ly is orthogonal to F and
so is (x + x') — (y + /) = (x - y) + (x' - /); as y + / e F and ky e F, this
shows that y + / = P_¥(x + x') and ly = P_F(Ax). By Pythagoras' theorem, we
have

(6.3.1.2) \\x\\² = \\P_F(x)\\² + ||^ - ?_F(x)||²

and this proves that ||P_F(x)|| < ||x||, hence (5.5.1) P_F is continuous and has
norm ^ 1 ; but as P_F(X) = x for x E F, we have ||P_F|| = 1 if F is not reduced
to 0. The definition of P_F implies that F' = Pf *(0) consists of the vectors x
orthogonal to F; as x = P_F(;c) + (x — P_F(x)) and x - P_F(x) e F^r for any
x e E, we have E = F + F'; moreover, if x e F n F', x is isotropic, hence



	120 VI HUBERT SPACES such that \\x — y\\ = d(x, F). The point y = P_F(x) is also the only point z e F such that x - z is orthogonal to F. The mapping x -> P_F(x) 0/ E onto F w /we<2r, continuous, and of norm 1 z/ F ^ {0}; its kernel F' = P~¹(Q) is the subspace orthogonal to ¥,andE is the topological direct sum (see Section (5.4)) o/F and F'. Finally, F w rte subspace orthogonal to F'. Let a = rf(je, F); by definition, there exists a sequence (y_n) of points of F such that lim \|\|x — j>J = a; we prove (y_n) is a Cauchy sequence. H-+OO Indeed, for any two points u, v of E, it follows from (6.1 .1) that (6.3.1 .1) \\u + v\\² + \\u - v\\² = 2(N\|² + \|\|i?\|\|²) hence \\y_m - yj² = 2(\|\|* - y_a\\² + \|\|x - jj²) - 4\|\|* - Ky_M + 7«)ll². But iC^m + 7J e F, hence \\x - %(y_m + y_n)\\² ^ a² ; therefore, if n_Q is such that for n ^ «₀ , \\x - y_n\\² < a² + e, we have, for m ^ w₀ and n ^ «₀ > Ibm - ^nii^? < ^4fi₃which proves our contention. As F is complete, the sequence (y_n) tends to a limit y e F, for which \|\|x - y\\ = d(x₉ F). Suppose y' e F is also such that \\x—y'\\=d(x,]?); using again (6.3.11), we obtain \|\|j>-/\|\|² = 4a²- 4 \\x - $(y + /)\|\|², and as %(y + /) e F, this implies \|\|j; - / 1\|² < 0, i.e. y^r - y. Let now z ^ 0 be any point of F, and write that \\x - ( y + Az)\|\|² > a²for any real scalar A ^ 0; this, by (6.1 .1 ), gives

	and this would yield a contradiction if we had $(x — y \ z) ^ 0, by a suitable choice of L Hence 3%(x — y \ z) = 0, and replacing z by /z (if E is a complex prehilbert space) shows that J(x — y \ z) = 0, hence (x — y \ z) = 0 in every case; in other words x — y is orthogonal to F. Let y' e F be such that x — / is orthogonal to F; then, for any z^O in F, we have \\x - (/ + z)\|\|² = \|\|x - / 1\|² + \|\|z\|\|² by Pythagoras' theorem, and this proves that / = y by the previous characterization of y. This last characterization of y = P_F(x) proves that P_F is linear, for if x — y and x' — y' are orthogonal to F, then Xx — ly is orthogonal to F and so is (x + x') — (y + /) = (x - y) + (x' - /); as y + / e F and ky e F, this shows that y + / = P_¥(x + x') and ly = P_F(Ax). By Pythagoras' theorem, we have (6.3.1.2) \\x\\² = \\P_F(x)\\² + \|\|^ - ?_F(x)\|\|² and this proves that \|\|P_F(x)\|\| < \|\|x\|\|, hence (5.5.1) P_F is continuous and has norm ^ 1 ; but as P_F(X) = x for x E F, we have \|\|P_F\|\| = 1 if F is not reduced to 0. The definition of P_F implies that F' = Pf (0) consists of the vectors x orthogonal to F; as x =* P_F(;c) + (x — P_F(x)) and x - P_F(x) e F^r for any x e E, we have E = F + F'; moreover, if x e F n F', x is isotropic, hence